Message #3495

From: Melinda Green <melinda@superliminal.com>
Subject: Re: [MC4D] Earthquake Puzzle
Date: Sun, 14 Aug 2016 18:42:00 -0700

How very cool, Roice!

I don’t quite understand how the twist highlighting works but I’m able
to sort of find the ones I want and solve a couple of random scrambling
twists.

You’ve probably already guessed what I’m going to which is whether this
puzzle could support the "Show as Skew" view. We’ve learned that solving
is better supported by the hyperbolic view but a 3D view would help in
understanding the topology of the puzzle. I stared a bit at this image
<http://math.ucr.edu/home/baez/pentacontihexahedron2.jpg> to try to
understand what’s going on. From your description it sounds like each
twist cuts one off the three struts of a red hub, turns one of those
arms around its cut, flipping the hub over and swapping the other two
struts. Is that correct? That seems to suggest that the scrambling
twists plus solving twists is always even. It also suggests there are
other types of possible twists. One of them seems like the most natural
one to me which twists a selected strut by 180 degrees, swapping the
hubs at each end. That one seems to be a "true" Big Chop-like deep cut
since it’s symmetric on both sides. Actually, it looks like there are
more than one way to do that too though the simple geometric rotation
seems the most natural.

The really neat thing about your new feature is that it works at a kind
of meta level by operating on the hubs and struts of high genus surfaces
similarly to how we’ve been twisting vertices and edges within them.
Heck, it looks like you could even create puzzles within puzzles where
you manipulate the structure like you are doing now while also allowing
users to twist the elements within the texture with a modifier key or
something. Does that make any sense?

Assuming I haven’t gone completely off into the weeds, I’d love to see
the {7,3} or other IRPs supported in this way. None of this is to
pressure you to implement anything but rather to try to understand what
this new puzzle means and where it could go.

Thanks for the wonderful new toy!
-Melinda

On 8/14/2016 1:47 PM, Roice Nelson roice3@gmail.com [4D_Cubing] wrote:
>
>
> Hi Hypercubers,
>
> I’ve got a new puzzle variant of the Klein Quartic surface for you
> that I’m excited to share. This puzzle was suggested to me by Arnaud
> Cheritat <http://www.math.univ-toulouse.fr/%7Echeritat/> at a workshop
> on illustrating mathematics, and uses a new kind of twisting. aside:
> Arnaud has made a beautiful applet
> <http://www.math.univ-toulouse.fr/%7Echeritat/AppletsDivers/Klein/> to
> explore the quartic.
>
> Rather than slice up the surface with circles that can be shrunk to a
> point, we slice it up with systolic (shortest length) geodesics.
> These geodesics cut the surface "around the horn" as Melinda has
> described in the past. To picture an analogous geodesic, think of a
> circle on a torus that can not be shrunk to a point, but which is
> shrunk as small as possible (see the beginning of this article
> <http://www.ams.org/notices/200803/tx080300374p.pdf>).
>
> Why call this the Earthquake? That was a term Arnaud was using, and
> it turns out quite descriptive when you see a portion of the surface
> shearing along a systole. It is even more appropriate because it is
> necessary to temporarily detach the surface from itself during the
> course of a twist. The surface remains connected along one systole
> (the movement near this slice reminds me of the "Big Chop" puzzle),
> but detaches along the other two systoles, swapping the material
> connected to each of them. The twist animation hopefully gives a
> flavor of the surface separating and reattaching to itself.
>
> I attempted to make it intuitive to control an earthquake twist, but
> note there are three ways to twist a set of systoles (6 if you count
> direction, but direction doesn’t affect state so it’s only a visual
> thing). Here’s a video <https://youtu.be/5w6-dD8YfoI> introducing the
> twisting. Here are a few images
> <https://goo.gl/photos/YvpdPvwNxzV8Z9CH6>, showing the puzzle pristine
> and scrambled.
>
> I have not tried to solve this puzzle yet. I hope it is a good
> challenge, though I wonder if the fact that twists result in 2-cycles
> of stickers might make it on the easy side. It certainly turned out
> to be a bear to implement!
>
> Grab the latest MagicTile
> <http://www.gravitation3d.com/magictile/downloads/MagicTile_v2.zip>,
> check it out, and give us some detailed notes of anything you discover
> while solving it!
>
> Cheers,
> Roice
>
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